\documentclass{article}

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\begin{document}
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	% dx or d[#1]
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	\begin{align*}
		&\ \int \sqrt{1 + x^2} \d
		\\
		&\ let\ \ x = \tan{t}
		\\
		= &\ I
		\\
		= &\ \int \sec{t} \d[\tan{t}]
		\\
		= &\ \sec{t} \tan{t} - \int \tan{t} \d[\sec{t}]
		\\
		= &\ \sec{t} \tan{t} - \int \tan^2{t} \sec{t} \d[t]
		\\
		= &\ \sec{t} \tan{t} - \int (\sec^2{t} - 1) \sec{t} \d[t]
		\\
		= &\ \sec{t} \tan{t} - \int \sec^3{t} \d[t] + \int \sec{t} \d[t]
		\\
		= &\ \sec{t} \tan{t} - I + \ln{|\sec{t} + \tan{t}|} + C
		\\
		\\
		\Rightarrow I = &\ \frac{1}{2} (\sec{t} \tan{t} + \ln{|\sec{t} + \tan{t}|}) + C
		\\
		= &\ \frac{1}{2} (x \sqrt{1 + x^2} + \ln{(x + \sqrt{1 + x^2})}) + C 
	\end{align*}
\end{document}
